Polytope of Type {24,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,36}*1728d
if this polytope has a name.
Group : SmallGroup(1728,5217)
Rank : 3
Schlafli Type : {24,36}
Number of vertices, edges, etc : 24, 432, 36
Order of s₀s₁s₂ : 72
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,36}*864a
   3-fold quotients : {8,36}*576b, {24,12}*576e
   4-fold quotients : {6,36}*432a, {12,18}*432a
   6-fold quotients : {4,36}*288a, {12,12}*288a
   8-fold quotients : {6,18}*216a
   9-fold quotients : {24,4}*192b, {8,12}*192b
   12-fold quotients : {2,36}*144, {4,18}*144a, {6,12}*144a, {12,6}*144a
   18-fold quotients : {4,12}*96a, {12,4}*96a
   24-fold quotients : {2,18}*72, {6,6}*72a
   27-fold quotients : {8,4}*64b
   36-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
   48-fold quotients : {2,9}*36
   54-fold quotients : {4,4}*32
   72-fold quotients : {2,6}*24, {6,2}*24
   108-fold quotients : {2,4}*16, {4,2}*16
   144-fold quotients : {2,3}*12, {3,2}*12
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)
( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)( 68, 98)
( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)( 76,106)
( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(109,136)(110,137)(111,138)
(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)
(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)
(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)
(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)
(186,189)(193,196)(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)
(212,215)(213,216)(217,325)(218,326)(219,327)(220,331)(221,332)(222,333)
(223,328)(224,329)(225,330)(226,334)(227,335)(228,336)(229,340)(230,341)
(231,342)(232,337)(233,338)(234,339)(235,343)(236,344)(237,345)(238,349)
(239,350)(240,351)(241,346)(242,347)(243,348)(244,352)(245,353)(246,354)
(247,358)(248,359)(249,360)(250,355)(251,356)(252,357)(253,361)(254,362)
(255,363)(256,367)(257,368)(258,369)(259,364)(260,365)(261,366)(262,370)
(263,371)(264,372)(265,376)(266,377)(267,378)(268,373)(269,374)(270,375)
(271,406)(272,407)(273,408)(274,412)(275,413)(276,414)(277,409)(278,410)
(279,411)(280,415)(281,416)(282,417)(283,421)(284,422)(285,423)(286,418)
(287,419)(288,420)(289,424)(290,425)(291,426)(292,430)(293,431)(294,432)
(295,427)(296,428)(297,429)(298,379)(299,380)(300,381)(301,385)(302,386)
(303,387)(304,382)(305,383)(306,384)(307,388)(308,389)(309,390)(310,394)
(311,395)(312,396)(313,391)(314,392)(315,393)(316,397)(317,398)(318,399)
(319,403)(320,404)(321,405)(322,400)(323,401)(324,402);;
s1 := (  1,220)(  2,222)(  3,221)(  4,217)(  5,219)(  6,218)(  7,223)(  8,225)
(  9,224)( 10,240)( 11,239)( 12,238)( 13,237)( 14,236)( 15,235)( 16,243)
( 17,242)( 18,241)( 19,231)( 20,230)( 21,229)( 22,228)( 23,227)( 24,226)
( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)( 32,246)
( 33,245)( 34,250)( 35,252)( 36,251)( 37,267)( 38,266)( 39,265)( 40,264)
( 41,263)( 42,262)( 43,270)( 44,269)( 45,268)( 46,258)( 47,257)( 48,256)
( 49,255)( 50,254)( 51,253)( 52,261)( 53,260)( 54,259)( 55,274)( 56,276)
( 57,275)( 58,271)( 59,273)( 60,272)( 61,277)( 62,279)( 63,278)( 64,294)
( 65,293)( 66,292)( 67,291)( 68,290)( 69,289)( 70,297)( 71,296)( 72,295)
( 73,285)( 74,284)( 75,283)( 76,282)( 77,281)( 78,280)( 79,288)( 80,287)
( 81,286)( 82,301)( 83,303)( 84,302)( 85,298)( 86,300)( 87,299)( 88,304)
( 89,306)( 90,305)( 91,321)( 92,320)( 93,319)( 94,318)( 95,317)( 96,316)
( 97,324)( 98,323)( 99,322)(100,312)(101,311)(102,310)(103,309)(104,308)
(105,307)(106,315)(107,314)(108,313)(109,355)(110,357)(111,356)(112,352)
(113,354)(114,353)(115,358)(116,360)(117,359)(118,375)(119,374)(120,373)
(121,372)(122,371)(123,370)(124,378)(125,377)(126,376)(127,366)(128,365)
(129,364)(130,363)(131,362)(132,361)(133,369)(134,368)(135,367)(136,328)
(137,330)(138,329)(139,325)(140,327)(141,326)(142,331)(143,333)(144,332)
(145,348)(146,347)(147,346)(148,345)(149,344)(150,343)(151,351)(152,350)
(153,349)(154,339)(155,338)(156,337)(157,336)(158,335)(159,334)(160,342)
(161,341)(162,340)(163,409)(164,411)(165,410)(166,406)(167,408)(168,407)
(169,412)(170,414)(171,413)(172,429)(173,428)(174,427)(175,426)(176,425)
(177,424)(178,432)(179,431)(180,430)(181,420)(182,419)(183,418)(184,417)
(185,416)(186,415)(187,423)(188,422)(189,421)(190,382)(191,384)(192,383)
(193,379)(194,381)(195,380)(196,385)(197,387)(198,386)(199,402)(200,401)
(201,400)(202,399)(203,398)(204,397)(205,405)(206,404)(207,403)(208,393)
(209,392)(210,391)(211,390)(212,389)(213,388)(214,396)(215,395)(216,394);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)(  8, 18)
(  9, 17)( 19, 21)( 22, 24)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 40)
( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 46, 48)( 49, 51)( 52, 54)
( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)( 62, 72)
( 63, 71)( 73, 75)( 76, 78)( 79, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)
( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(100,102)(103,105)(106,108)
(109,145)(110,147)(111,146)(112,148)(113,150)(114,149)(115,151)(116,153)
(117,152)(118,136)(119,138)(120,137)(121,139)(122,141)(123,140)(124,142)
(125,144)(126,143)(127,156)(128,155)(129,154)(130,159)(131,158)(132,157)
(133,162)(134,161)(135,160)(163,199)(164,201)(165,200)(166,202)(167,204)
(168,203)(169,205)(170,207)(171,206)(172,190)(173,192)(174,191)(175,193)
(176,195)(177,194)(178,196)(179,198)(180,197)(181,210)(182,209)(183,208)
(184,213)(185,212)(186,211)(187,216)(188,215)(189,214)(217,280)(218,282)
(219,281)(220,283)(221,285)(222,284)(223,286)(224,288)(225,287)(226,271)
(227,273)(228,272)(229,274)(230,276)(231,275)(232,277)(233,279)(234,278)
(235,291)(236,290)(237,289)(238,294)(239,293)(240,292)(241,297)(242,296)
(243,295)(244,307)(245,309)(246,308)(247,310)(248,312)(249,311)(250,313)
(251,315)(252,314)(253,298)(254,300)(255,299)(256,301)(257,303)(258,302)
(259,304)(260,306)(261,305)(262,318)(263,317)(264,316)(265,321)(266,320)
(267,319)(268,324)(269,323)(270,322)(325,415)(326,417)(327,416)(328,418)
(329,420)(330,419)(331,421)(332,423)(333,422)(334,406)(335,408)(336,407)
(337,409)(338,411)(339,410)(340,412)(341,414)(342,413)(343,426)(344,425)
(345,424)(346,429)(347,428)(348,427)(349,432)(350,431)(351,430)(352,388)
(353,390)(354,389)(355,391)(356,393)(357,392)(358,394)(359,396)(360,395)
(361,379)(362,381)(363,380)(364,382)(365,384)(366,383)(367,385)(368,387)
(369,386)(370,399)(371,398)(372,397)(373,402)(374,401)(375,400)(376,405)
(377,404)(378,403);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(432)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)
( 60, 90)( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)
( 68, 98)( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)
( 76,106)( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(109,136)(110,137)
(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)
(119,146)(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)
(127,154)(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)
(135,159)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)
(185,188)(186,189)(193,196)(194,197)(195,198)(202,205)(203,206)(204,207)
(211,214)(212,215)(213,216)(217,325)(218,326)(219,327)(220,331)(221,332)
(222,333)(223,328)(224,329)(225,330)(226,334)(227,335)(228,336)(229,340)
(230,341)(231,342)(232,337)(233,338)(234,339)(235,343)(236,344)(237,345)
(238,349)(239,350)(240,351)(241,346)(242,347)(243,348)(244,352)(245,353)
(246,354)(247,358)(248,359)(249,360)(250,355)(251,356)(252,357)(253,361)
(254,362)(255,363)(256,367)(257,368)(258,369)(259,364)(260,365)(261,366)
(262,370)(263,371)(264,372)(265,376)(266,377)(267,378)(268,373)(269,374)
(270,375)(271,406)(272,407)(273,408)(274,412)(275,413)(276,414)(277,409)
(278,410)(279,411)(280,415)(281,416)(282,417)(283,421)(284,422)(285,423)
(286,418)(287,419)(288,420)(289,424)(290,425)(291,426)(292,430)(293,431)
(294,432)(295,427)(296,428)(297,429)(298,379)(299,380)(300,381)(301,385)
(302,386)(303,387)(304,382)(305,383)(306,384)(307,388)(308,389)(309,390)
(310,394)(311,395)(312,396)(313,391)(314,392)(315,393)(316,397)(317,398)
(318,399)(319,403)(320,404)(321,405)(322,400)(323,401)(324,402);
s1 := Sym(432)!(  1,220)(  2,222)(  3,221)(  4,217)(  5,219)(  6,218)(  7,223)
(  8,225)(  9,224)( 10,240)( 11,239)( 12,238)( 13,237)( 14,236)( 15,235)
( 16,243)( 17,242)( 18,241)( 19,231)( 20,230)( 21,229)( 22,228)( 23,227)
( 24,226)( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)
( 32,246)( 33,245)( 34,250)( 35,252)( 36,251)( 37,267)( 38,266)( 39,265)
( 40,264)( 41,263)( 42,262)( 43,270)( 44,269)( 45,268)( 46,258)( 47,257)
( 48,256)( 49,255)( 50,254)( 51,253)( 52,261)( 53,260)( 54,259)( 55,274)
( 56,276)( 57,275)( 58,271)( 59,273)( 60,272)( 61,277)( 62,279)( 63,278)
( 64,294)( 65,293)( 66,292)( 67,291)( 68,290)( 69,289)( 70,297)( 71,296)
( 72,295)( 73,285)( 74,284)( 75,283)( 76,282)( 77,281)( 78,280)( 79,288)
( 80,287)( 81,286)( 82,301)( 83,303)( 84,302)( 85,298)( 86,300)( 87,299)
( 88,304)( 89,306)( 90,305)( 91,321)( 92,320)( 93,319)( 94,318)( 95,317)
( 96,316)( 97,324)( 98,323)( 99,322)(100,312)(101,311)(102,310)(103,309)
(104,308)(105,307)(106,315)(107,314)(108,313)(109,355)(110,357)(111,356)
(112,352)(113,354)(114,353)(115,358)(116,360)(117,359)(118,375)(119,374)
(120,373)(121,372)(122,371)(123,370)(124,378)(125,377)(126,376)(127,366)
(128,365)(129,364)(130,363)(131,362)(132,361)(133,369)(134,368)(135,367)
(136,328)(137,330)(138,329)(139,325)(140,327)(141,326)(142,331)(143,333)
(144,332)(145,348)(146,347)(147,346)(148,345)(149,344)(150,343)(151,351)
(152,350)(153,349)(154,339)(155,338)(156,337)(157,336)(158,335)(159,334)
(160,342)(161,341)(162,340)(163,409)(164,411)(165,410)(166,406)(167,408)
(168,407)(169,412)(170,414)(171,413)(172,429)(173,428)(174,427)(175,426)
(176,425)(177,424)(178,432)(179,431)(180,430)(181,420)(182,419)(183,418)
(184,417)(185,416)(186,415)(187,423)(188,422)(189,421)(190,382)(191,384)
(192,383)(193,379)(194,381)(195,380)(196,385)(197,387)(198,386)(199,402)
(200,401)(201,400)(202,399)(203,398)(204,397)(205,405)(206,404)(207,403)
(208,393)(209,392)(210,391)(211,390)(212,389)(213,388)(214,396)(215,395)
(216,394);
s2 := Sym(432)!(  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)
(  8, 18)(  9, 17)( 19, 21)( 22, 24)( 25, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 40)( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 46, 48)( 49, 51)
( 52, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)
( 62, 72)( 63, 71)( 73, 75)( 76, 78)( 79, 81)( 82, 91)( 83, 93)( 84, 92)
( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(100,102)(103,105)
(106,108)(109,145)(110,147)(111,146)(112,148)(113,150)(114,149)(115,151)
(116,153)(117,152)(118,136)(119,138)(120,137)(121,139)(122,141)(123,140)
(124,142)(125,144)(126,143)(127,156)(128,155)(129,154)(130,159)(131,158)
(132,157)(133,162)(134,161)(135,160)(163,199)(164,201)(165,200)(166,202)
(167,204)(168,203)(169,205)(170,207)(171,206)(172,190)(173,192)(174,191)
(175,193)(176,195)(177,194)(178,196)(179,198)(180,197)(181,210)(182,209)
(183,208)(184,213)(185,212)(186,211)(187,216)(188,215)(189,214)(217,280)
(218,282)(219,281)(220,283)(221,285)(222,284)(223,286)(224,288)(225,287)
(226,271)(227,273)(228,272)(229,274)(230,276)(231,275)(232,277)(233,279)
(234,278)(235,291)(236,290)(237,289)(238,294)(239,293)(240,292)(241,297)
(242,296)(243,295)(244,307)(245,309)(246,308)(247,310)(248,312)(249,311)
(250,313)(251,315)(252,314)(253,298)(254,300)(255,299)(256,301)(257,303)
(258,302)(259,304)(260,306)(261,305)(262,318)(263,317)(264,316)(265,321)
(266,320)(267,319)(268,324)(269,323)(270,322)(325,415)(326,417)(327,416)
(328,418)(329,420)(330,419)(331,421)(332,423)(333,422)(334,406)(335,408)
(336,407)(337,409)(338,411)(339,410)(340,412)(341,414)(342,413)(343,426)
(344,425)(345,424)(346,429)(347,428)(348,427)(349,432)(350,431)(351,430)
(352,388)(353,390)(354,389)(355,391)(356,393)(357,392)(358,394)(359,396)
(360,395)(361,379)(362,381)(363,380)(364,382)(365,384)(366,383)(367,385)
(368,387)(369,386)(370,399)(371,398)(372,397)(373,402)(374,401)(375,400)
(376,405)(377,404)(378,403);
poly := sub<Sym(432)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope